Light can be characterized as a wave having frequency, amplitude, and phase. Because these waves oscillate too quickly to be detected directly, we can only detect their intensity, not their phase. Phase, however, carries information about an object's optical density and thickness. Transparent objects, for example, do not change the amplitude of the light passing through them, but introduce phase delays due to regions of higher optical density (refractive index). Optical density is related to physical density, and thus phase imaging can give distributions of pressure, temperature, humidity, and other material properties. The phase of reflected waves can also be used to derive information about the topology, or surface profile, of the object from which the waves were reflected.
In phase contrast imaging, a lens separates the dc component of light transmitted through a transparent or translucent object from components at higher spatial frequencies. A phase mask shifts the phase of the dc component with respect to components of the wavefront at higher spatial frequencies. Fourier transforming the wavefront causes the phase-shifted dc component to interfere with the higher-frequency components, resulting in an image whose contrast depends on the object's optical density. Although phase-contrast imaging is relatively simple, it provides only qualitative information about the optical density of the object being imaged.
Most quantitative phase imaging methods involve interferometry, which usually requires coherent illumination and stable reference beams. For example, interferometry can be used to produce a fringe pattern whose period depends on the slope of a surface relative to a reference plane. Unfortunately, producing the fringe pattern requires illuminating the surface with coherent light and stabilizing the path lengths of the reference and object beams to within a fraction of the illumination wavelength. The fringe pattern must also be unwrapped, which is difficult to do for two-dimensional fringe patterns or phase profiles with steep phase gradients or significant noise.
Noninterferometric phase measurement techniques are better-suited for use in adaptive optical systems because they do not need stabilized reference beams. For example, a Shack-Hartmann sensor measures wavefront error (phase difference) by transforming spatial phase variations in an incident wavefront into spatial displacements. A lenslet array transforms the wavefront into an array of spots, where the transverse position of each spot depends on the phase of the portion of the wavefront sampled by the corresponding lenslet. Unfortunately, Shack-Hartmann sensors tend to have poor spatial resolution because the number of lenslets in the lenslet array must be several times smaller than the number of pixels used to detect the shifted spots.
The Transport of Intensity Equation (TIE) is a wave-optical phase retrieval technique that involves measuring the derivative of intensity along the optical axis by taking images at varying depths, then solving for the phase. Typically, the TIE approach uses three images: an in-focus image, an under-focused image, and an over-focused image, where the under- and over-focused images are symmetric about the in-focus image. The out-of-focus images are used to estimate the derivative of the intensity in the direction of propagation, which, in turn, can be used to solve the TIE for the phase, provided that the out-of-focus images are within the small defocus limit. For more on the TIE approach, see M. Beleggia et al., “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102:37-49 (2004), incorporated herein by reference in its entirety. Although TIE does not require a reference beam, it does require a mechanically stable platform and multiple sequential measurements, so it is not suitable for use in real-time measurement or feedback loops, such as those used in adaptive optical systems.